Ratios and proportions
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A ratio is a comparison of two quantities in a simplified form.
In Quantitative Reasoning, you may see a ratio written as:
- numbers separated by a colon, such as 4:3 or 1:2:5.
- a fraction, whether written with a slash or a proper fraction bar, such as 4/3 or .
A ratio can also be described in words, with ‘to’ in place of the colon, slash or fraction bar. Sometimes, in a full sentence, another word like ‘for’ could take the place of ‘to’.
Most commonly in the UCAT, a ratio will have two parts, like a conventional fraction.
Less commonly in the UCAT, a ratio will have three parts.
In this case the ratio cannot be directly written as a fraction.
However, you can work with just two parts as a fraction if you are careful with the maths.
If you must form a two-part ratio in Quantitative Reasoning:
- Take a moment to pick the correct values from the table, chart or graph
- Be careful to accurately assign the numerator and the denominator. You are likely to select a wrong answer trap if you mistakenly solve for the reciprocal of the correct answer.
You can then reduce the ratio in your head or divide it out using the calculator. Your approach will depend on time pressures and also your comfort level.
If you spot a common factor in both sides of a ratio, you may wish to reduce before you divide.
- You can reduce both sides of a ratio in colon form just as you would reduce a fraction.
- As you practise, you may want to reduce ratios on scrap paper to simulate using the notebook for rough working on Test Day.
- Focus on simple factors such as small primes like 2, 3 and 5, but you may quickly see composite factors such as 6, 8, 10, 25 or multiples of 10.
This approach works best if you can reduce one side of the ratio to 1 or to a very small value. Otherwise, you may prefer to simply divide it out using the calculator.
Many UCAT students prefer to divide out a ratio using the onscreen calculator.
This can be a useful way to save time when it isn’t obvious how to simplify.
A helpful benchmark for any new ratio is to eyeball its value in comparison to 1.
If the numerator is less than the denominator we know the ratio will be between 0 and 1.
If the numerator is more than the denominator we know the ratio will be more than 1.
Walkthrough question
Ratios
Let's try a question involving ratios.
A ratio can be written side-to-side using a colon, or it can be expressed as a fraction.
As a result, you can convert directly between:
- ratios
- fractions
- decimals
- percentages
Be ready to convert fluidly and quickly as needed, depending on the specific maths required to calculate efficiently or to check the answer choices.
You may see the word ‘proportion’ in Quantitative Reasoning.
When this happens it is important we consider its two meanings.
1. The everyday sense of a part of a whole, similar to a fraction or percentage.
2. The technical maths sense of two equivalent ratios or fractions
If a QR question asks you to solve for a proportion but the answers are fractions, ratios or percentages, it is very likely a proportion in the everyday sense.
Sometimes, you must use two equivalent ratios or fractions to solve for an unknown value in one part of the two ratios or fractions. You would normally cross-multiply to solve for the variable.
This approach is similar to how proportions are taught in maths classes. However, you can often save a step by mentally setting up the algebra so you can directly calculate the unknown in the calculator. This requires you to rearrange the fractions to isolate the variable and thereby pre-plan the necessary calculations to do them in one go.
It is always fine to use the notebook for any rough working. But if you train your brain to set up before you solve, you can go right to the calculator with speed and accuracy.
Walkthrough question
Proportions
Let's try using proportions in this QR question.
A part-to-part ratio compares two different parts of the data.
Ensure that both parts in a part-to-part ratio are in the same units. Otherwise, the comparison is inaccurate. This means you may need to convert units as a first step.
You can work with part-to-part ratios in any format. It may be simpler to express the comparison as a fraction, decimal or percentage, depending on the maths involved.
However, the UCAT will normally give ratios in simplest form in the answer choices. Thus, you may prefer to reduce common factors as you work. This may also help to minimise maths as you may find one side of the ratio reduces to 1 or a small factor like 2 or 3.
You will often save time in Quantitative Reasoning if you compare a ratio to a benchmark fraction to estimate the ratio’s value.
This is a quick eyeball comparison to see if the ratio is greater or less than the benchmark fraction.
Remember that when you compare two fractions or ratios:
- If two fractions have the same denominator but different numerators, the fraction with the higher numerator is greater than the other fraction. This reflects that the greater fraction has more equally-sized portions.
- If two fractions have the same numerator but different denominators, the fraction with the lower denominator is greater than the other fraction. This reflects that the greater fraction has an equal number of larger portions.
You can use any easily recognised fraction as a benchmark fraction. In most cases, these will be fractions between 0 and 1. Bear in mind it is often simpler to ‘see’ the value if you think in terms of the decimal or percentage equivalent.
The simplest options are one-half and the thirds, quarters, fifths, sixths and tenths between 0 and 1. Consider how this creates a sequence of benchmark values between 0 and 1:
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You can expand your benchmark fractions to increase accuracy in QR if you:
- memorise the decimal equivalents for sevenths and eighths between 0 and 1.
- learn the simple rules to recall the repeating decimal for ninths and elevenths between 0 and 1.
Walkthrough exercise
Decimal equivalents - ninths and elevenths
Take a look at how to convert to fractions with ninths and elevenths.
Walkthrough question
Ninths and elevenths
Try using benchmark fractions to answer this QR question on ratios.
Walkthrough exercise
Benchmark fractions between 0 and 1
Take a look at this list of expanded benchmark fractions.
Be prepared to quickly and brutally eliminate answer choices where the ratio is far too big or too small compared to your calculation or estimate.
Sometimes, two or three wrong answers are quite obviously too big or too small.
Walkthrough question
Part-to-part ratios
Now let's attempt a question involving part-to-part ratios.
A part-to-whole ratio compares part of the data to the whole amount.
You may need to add up a row or column from the table, or otherwise find the sum that you will use as the whole.
Sometimes, you must add up two or more portions from the data into a single part, which you then compare to the whole amount in a part-to-whole ratio.
A percentage is the number of parts per hundred.
Since it compares a part to a whole, a percentage is a proportion and thus also a part-to-whole ratio.
We can think of the percentage formula as:
Watch out for wording variations that can make it tough to translate words into maths.
There are many ways the UCAT could ask you to work with part-to-whole ratios, which could include starting or ending in percentage or decimal form.
The key is whether you must find parts per hundred or one portion as a quantity of the total amount.
This maths concept can be built into many real-world contexts, from test scores to population figures to market shares. Use the maths logic to figure out how to set up and solve.
Sometimes, you must manipulate the parts before you form your ratio and calculate. You might need to:
- add up the portions in the initial ratio to find the total number of portions. This applies in both two-part and three-part ratios.
- add up portions to go on one side of your new ratio, whether a part-to-part or part-to-whole ratio.
- multiply both sides of a ratio by a common factor in order to compare two different ratios with a single common part.
Walkthrough example
Forming a ratio
Let's look at how to form new part-to-part or part-to-whole ratios.
Look for key words and phrases in the question stem to decide what type of ratio is required.
Let’s assume a group of people includes three distinct subgroups: x, y and z. Some common structures:
- ‘What is the ratio of x to y?’: solve for a part-to-part ratio.
- ‘What proportion of the people are x or y?’: solve for a part-to-whole ratio, which is x + y divided by x + y + z. Sum up the part, sum up the whole, then divide.
- ‘What is the ratio of y and z to x?’: a more challenging part-to-part ratio, as you must sum up y + z and divide by x.
Watch out for two essential key words:
- ‘to’ comes just before the denominator or the right side of the ratio in colon form.
- ‘of’ immediately precedes the numerator or the left side of the ratio in colon form when used in conjunction with ‘to’.
- ‘of’ used without ‘to’ could indicate the whole group in a proportion or a part-to-whole ratio.
Walkthrough question
Part-to-whole ratios
Take a look at this question involving part-to-whole ratios.
You may need to simplify a ratio either as an initial step before further calculations, or as a final step to try and get your value in a format that is closer to the answer choices.
Some common techniques:
- Reduce a simple, obvious factor from both sides, such as 2, 5 or 10.
- Reduce all common factors other than 1 from both sides, though it can be slower and frustrating to do so under timed conditions.
- Reduce the greatest common factor from both sides, though you may make an error if you try to reduce in your head using this approach.
- Look for larger factors – but not necessarily the greatest common factor – that you can reduce from both sides. If there are very large values, you can sometimes reduce a composite factor such as 25, 40 or 60.
As a backup, you can always divide out the ratio in the calculator and compare decimal values. This can also work well if you are comfortable with benchmark fractions and their decimal equivalents.
Try to train yourself to avoid extra steps, which can cause you to waste time in a ratio or proportion question.
Remember that a ratio is a fraction, so you can always use a fraction, decimal or percentage – whatever will minimise maths and let you get to the answer faster.
Walkthrough question
Proportion challenges
Now attempt a difficult question with a proportion.